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This chapter returns to the juncture in Chapter II where it was assumed, temporarily, that the historical and continuing inter-relations of pure mathematics with natural science are enough to warrant the Second Philosopher in regarding set theory as a body of truths. This assumption led to Thin Realism. Without it, the Second Philosopher is led instead to Arealism, the view that set theory is a highly effective practice of developing a theory to do a range of important mathematical jobs, but that it isn't in the business of uncovering truths or describing existing objects. Despite their stark...

This chapter returns to the juncture in Chapter II where it was assumed, temporarily, that the historical and continuing inter-relations of pure mathematics with natural science are enough to warrant the Second Philosopher in regarding set theory as a body of truths. This assumption led to Thin Realism. Without it, the Second Philosopher is led instead to Arealism, the view that set theory is a highly effective practice of developing a theory to do a range of important mathematical jobs, but that it isn't in the business of uncovering truths or describing existing objects. Despite their stark disagreement over truth and existence, the Thin Realist and the Arealist pursue set theory using exactly the same methods, constrained by exactly the same objective facts of mathematical depth. The conviction that there is a substantive difference between the two positions rests on dubious pre-conceptions about the determinacy of such notions as ‘true’ and ‘exist’.